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For the statistical functional $T(F)=\mu ={\mathbb{E}}_{F}X$, assuming ${\mathbb{E}}_F|X|<\infty$, the contaminated mean is exactlyThus the influence function of the mean is $x-\mu$, and its gross-error sensitivity is infinite on an unbounded sample space. The linear approximation is exact here: any fixed positive contamination fraction allows an arbitrarily large change by moving $x$ far enough. No finite second moment is needed for this derivative, although a finite variance is needed for the usual central limit theorem for the sample mean.

For comparison, the influence function of the sample median iswhen the median $m$ is unique and the probability density function $f$ is positive and continuous at $m$. The sign function records which side of $m$ receives the contaminating mass. The influence function of a quantile supplies the short derivation. With the generalized-inverse quantile convention, contamination exactly at $x=m$ leaves the median at $m$, giving derivative zero there. If $f(m)$ is very small, this bounded influence function can still be large; if $f(m)=0$, the displayed formula is unavailable.

For $F=N(\mu ,{\sigma }^2)$, put $z=(x-\mu )/\sigma$. The influence function of the mean, influence function of the sample median, and influence function of a Huber location estimator with fixed $s=\sigma$ are respectivelyThe first grows without bound, the second jumps at the median, and the third clips continuously. For $k=1.345$, $2\Phi (k)-1\simeq 0.821375$ and the normalized Huber tail is about $1.63750\sigma$. This illustrates the optimal bounded influence function for normal location without asserting that this cutoff is optimal for every contamination model.

Both panels are theoretical normal distributions, with no sampled or observed data. Doubling $\sigma$ doubles the vertical influence function scale and the horizontal clipping distance. Changing $\mu$ translates the curves. The isolated dot marks the actual median derivative at the center; connecting its two tails by a vertical segment would misrepresent that value. Bounded influence does not mean every small residual has less effect than on the mean: the central Huber slope is $1/(2\Phi (k)-1)>1$, a price paid for reducing sensitivity in the tails.